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Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
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Reconstructing an Affine Variety from its Coordinate Ring

I'm trying to understand the construction often written as $V=\operatorname{Spec}(R)$, where $R$ is a finitely generated $\mathbb C$-algebra with no nonzero nilpotents. At first glance, the notation $\operatorname{Spec}(R)$ is introduced just as the…
Christoph
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Rank one sheaves and ideal sheaves

For a coherent sheaf $\mathcal F$ on a smooth irreducible projective variety $X/k$, it makes sense to define the rank $\textrm{rk }\mathcal F$ as the rank of the vector bundle $\mathcal F|_U$, where $U$ is the open subset of $X$ where $\mathcal F$…
Brenin
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The "simplest" Hodge numbers of normal varieties

I realised yesterday (to my slight embarrassment) that I don't understand how the simplest Hodge numbers $h^{0,q}(X)$ and $h^{q,0}(X)$ behave when $X$ is singular. But I am sure the M.SE community can set me straight! Let's stick to normal…
user64687
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Using Valuative Criterion for Properness

How do I use the valuative criterion for properness: "Let $f: X \rightarrow Y$ be a morphism of finite type of noetherian schemes. Then $f$ is proper if and only if for all discrete valuation rings $R$ with fields of fractions $K$ and for any…
Gazerun
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field of rational functions of a projective variety equal to that of an affine variety

Let $Y \subset \mathbb{P}^n$ be a projective variety and let $U_i$ be the open set $x_i \neq 0$. Let $\phi_i : U_i \rightarrow \mathbb{A}^n$ be the isomorphism of varieties, defined e.g. in Hartshorne p. 10, that takes a point $P=(a_0,\cdots,a_n)…
Manos
  • 25,833
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intuitive interpretation of dimension of an affine variety

On the technical level i do understand what is the dimension of an affine variety and the connection of the definition with the Krull dimension or rings.However, what is the intuitive interpretation of the dimension of an affine variety? What does…
Manos
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Different definitions of rational mappings.

In the book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by David A. Cox, John Little, Donal O'Shea, the rational mapping is defined as follows. In the book "Algebraic Geometry: A…
LJR
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What's the relation between cohomology and unramified Galois covering of curves

The following statement in a paper puzzles me: "We may view $H^1(X(N), \mathbb{Z}/\ell\mathbb{Z})$ as classifying unramified Galois coverings of $X(N)$ with structure group $\mathbb{Z}/\ell\mathbb{Z}$." Here $X(N)$ is the usual modular curve of…
Jiangwei Xue
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Does an invertible sheaf always have global generating sections?

Theorem 7.1 in chapter 2 of Hartshorne's text says that invertible sheaves on a scheme $X$ together with its given global generating sections correspond to morphisms from $X$ to $\mathbb{P}_A^n$ (here $A$ is some fixed ring). I wonder if given any…
mr.bigproblem
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Curve over ring, covered by two affines

The reference is http://www.math.columbia.edu/~masdeu/files/notes/FallSeminar.pdf, page 9: Let now $C/R$ be a curve over a noetherian ring $R$; this means that $C$ is smooth, connected, integral, proper and of relative dimension $1$ over $R$. For…
user84471
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effective Cartier divisor is trivial

Given a schema $X/k$ with $H^0(X,\mathcal{O}_X^\times) = k^\times$ and an effective Cartier divisor $D \geq 0$ such that $\mathcal{O}(D) = O_X$, why is necessarily $D = 0$? I tried to apply the long exact cohomology sequence to $1 \to…
user5262
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On an exercise from Hartshone's Algebraic Geometry, Ch I sect 4

My question is about the Ex. 4.9 page 31 in the book Algebraic Geometry by Robin Hartshone. Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$, with $n\geq r+2$. Show that for suitable choice of $P\notin X$, and a linear…
fiorerb
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Graphs of transcendental functions are not algebraic varieties

I am trying to show that the zero set of $y - e^x$ is not an affine algebraic variety in $\mathbb{A}^2$. My idea has been to show that any polynomial vanishing on the zeros of $y - e^x$ must vanish on the whole plane, since then the nullstellensatz…
James Miller
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Numerically equivalent divisors on a surface

What is an example of divisors on a surface which are numerically equivalent but not linearly equivalent? Can a curve be numerically equivalent to 0? I don't have a good intuition for this. In projective space, any two curves intersect. In some less…
user64480
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Is the Picard Group countable?

Is the Picard group of a (smooth, projective) variety always countable? This seems likely but I have no idea if it's true. If so, is the Picard group necessarily finitely generated?
user64480
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